We prove that there exists a positive, explicit function F(k,E) such that, for any group G admitting a k-acylindrical splitting and any generating set S of G with Ent(G,S), we have ∣S∣≤F(k,E). We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, D-quasiconvexk-malnormal amalgamated products acting on δ-hyperbolic spaces or on CAT(0)-spaces with entropy bounded by E. A number of finiteness results for interesting families of Riemannian or metric spaces with bounded entropy and diameter also follow: Riemannian 2-orbifolds, non-geometric 3-manifolds, higher dimensional graph manifolds and cusp-decomposable manifolds, ramified coverings and, more generally, CAT(0)-groups with negatively curved splittings.
Filippo Cerocchi; Andrea Sambusetti. Entropy and finiteness of groups with acylindrical splittings. Groups, geometry, and dynamics, Tome 15 (2021) no. 3, pp. 755-799. doi: 10.4171/ggd/611
@article{10_4171_ggd_611,
author = {Filippo Cerocchi and Andrea Sambusetti},
title = {Entropy and finiteness of groups with acylindrical splittings},
journal = {Groups, geometry, and dynamics},
pages = {755--799},
year = {2021},
volume = {15},
number = {3},
doi = {10.4171/ggd/611},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/611/}
}
TY - JOUR
AU - Filippo Cerocchi
AU - Andrea Sambusetti
TI - Entropy and finiteness of groups with acylindrical splittings
JO - Groups, geometry, and dynamics
PY - 2021
SP - 755
EP - 799
VL - 15
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/611/
DO - 10.4171/ggd/611
ID - 10_4171_ggd_611
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%0 Journal Article
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%P 755-799
%V 15
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%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/611/
%R 10.4171/ggd/611
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